Optimal. Leaf size=116 \[ \frac {3 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{8 \left (a+b x^2\right )}+\frac {(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {424, 393, 211}
\begin {gather*} \frac {3 x \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right )}{8 \left (a+b x^2\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right )}{8 a^{5/2} b^{5/2}}+\frac {x \left (c+d x^2\right ) (b c-a d)}{4 a b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^3} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac {\int \frac {c (3 b c+a d)+d (b c+3 a d) x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {3 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{8 \left (a+b x^2\right )}+\frac {(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=\frac {3 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{8 \left (a+b x^2\right )}+\frac {(b c-a d) x \left (c+d x^2\right )}{4 a b \left (a+b x^2\right )^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 124, normalized size = 1.07 \begin {gather*} \frac {x \left (-3 a^3 d^2+3 b^3 c^2 x^2+a b^2 c \left (5 c+2 d x^2\right )-a^2 b d \left (2 c+5 d x^2\right )\right )}{8 a^2 b^2 \left (a+b x^2\right )^2}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 124, normalized size = 1.07
method | result | size |
default | \(\frac {-\frac {\left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) x^{3}}{8 a^{2} b}-\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) x}{8 b^{2} a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} b^{2} \sqrt {a b}}\) | \(124\) |
risch | \(\frac {-\frac {\left (5 a^{2} d^{2}-2 a b c d -3 b^{2} c^{2}\right ) x^{3}}{8 a^{2} b}-\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) x}{8 b^{2} a}}{\left (b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) d^{2}}{16 \sqrt {-a b}\, b^{2}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c d}{8 \sqrt {-a b}\, b a}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) c^{2}}{16 \sqrt {-a b}\, a^{2}}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) d^{2}}{16 \sqrt {-a b}\, b^{2}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c d}{8 \sqrt {-a b}\, b a}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) c^{2}}{16 \sqrt {-a b}\, a^{2}}\) | \(236\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 138, normalized size = 1.19 \begin {gather*} \frac {{\left (3 \, b^{3} c^{2} + 2 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} x}{8 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} + \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 213 vs.
\(2 (102) = 204\).
time = 0.48, size = 449, normalized size = 3.87 \begin {gather*} \left [\frac {2 \, {\left (3 \, a b^{4} c^{2} + 2 \, a^{2} b^{3} c d - 5 \, a^{3} b^{2} d^{2}\right )} x^{3} - {\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x}{16 \, {\left (a^{3} b^{5} x^{4} + 2 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}, \frac {{\left (3 \, a b^{4} c^{2} + 2 \, a^{2} b^{3} c d - 5 \, a^{3} b^{2} d^{2}\right )} x^{3} + {\left (3 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + 3 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d - 3 \, a^{4} b d^{2}\right )} x}{8 \, {\left (a^{3} b^{5} x^{4} + 2 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs.
\(2 (110) = 220\).
time = 0.54, size = 223, normalized size = 1.92 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b^{5}}} \cdot \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- a^{3} b^{2} \sqrt {- \frac {1}{a^{5} b^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{5}}} \cdot \left (3 a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2}\right ) \log {\left (a^{3} b^{2} \sqrt {- \frac {1}{a^{5} b^{5}}} + x \right )}}{16} + \frac {x^{3} \left (- 5 a^{2} b d^{2} + 2 a b^{2} c d + 3 b^{3} c^{2}\right ) + x \left (- 3 a^{3} d^{2} - 2 a^{2} b c d + 5 a b^{2} c^{2}\right )}{8 a^{4} b^{2} + 16 a^{3} b^{3} x^{2} + 8 a^{2} b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.32, size = 126, normalized size = 1.09 \begin {gather*} \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{2}} + \frac {3 \, b^{3} c^{2} x^{3} + 2 \, a b^{2} c d x^{3} - 5 \, a^{2} b d^{2} x^{3} + 5 \, a b^{2} c^{2} x - 2 \, a^{2} b c d x - 3 \, a^{3} d^{2} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.02, size = 130, normalized size = 1.12 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^{5/2}\,b^{5/2}}-\frac {\frac {x\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d-5\,b^2\,c^2\right )}{8\,a\,b^2}-\frac {x^3\,\left (-5\,a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^2\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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